trigonometric ratios in the unit circle
DESCRIPTION
Trigonometric Ratios in the Unit Circle. Warm-up (2 m). Sketch the following radian measures:. Trigonometric Ratios in the Unit Circle. The unit circle has a radius of 1. Quadrant II. Quadrant I. x is y is. x is y is. Quadrant III. Quadrant IV. x is y is. x is y is. - PowerPoint PPT PresentationTRANSCRIPT
Trigonometric Ratios in the Unit Circle
Warm-up (2 m)
1. Sketch the following radian measures:
6π17
65
Trigonometric Ratios in the Unit Circle The unit circle has a
radius of 1
θtanxyθtan
θcosrxθcos
θsinryθsin
x is
y is
x is
y is
x is
y is
x is
y is
Quadrant IQuadrant II
Quadrant III Quadrant IV
“All Students Take Calculus”AS
CT
all ratios are positive
sine is positive
tangent is positive
cosine is positive
cosecant is positive
cotangent is positive
secant is positive
Example:Trigonometric Ratio
Sine
Cosine
Tangent
5π
Example: 18π31
Trigonometric Ratio
Sine
Cosine
Tangent
Your Turn: Complete problems 1 - 3
Sketching Negative Radians and/or Multiple Revolutions
1. Whenever the angle is less than 0 or more than 2 pi, solve for the coterminal angle between 0 and 2 pi
2. Sketch the coterminal angle
Example #3:3π5
Trigonometric Ratio
Sine
Cosine
Tangent
Example #4: 5π23
Trigonometric Ratio
Sine
Cosine
Tangent
Your Turn: Complete practice problems 4 – 7
Reminder: Special Right Triangles
23
21 2
2
30°
60°
45°
45°
11
22
30° – 60° – 90° 45° – 45° – 90°
Investigation! Fit the paper triangles onto the picture below.
The side with the * must be on the x-axis. Use the paper triangles to determine the coordinates of the three points.
Special Right Triangles & the Unit Circle
Special Right Triangles & the Unit Circle: 30°- 60°
30°- 60°
45° or 4π
45° or 4π
Summarizing Questions1. In which quadrants is tangent positive?
Why?2. In which quadrants is cosecant negative?
Why?3. How do I sketch negative angles?4. How can I sketch angles with multiple
revolutions?5. What are some ways of remembering the
radian measures of the Unit Circle?6. How do we get the coordinates for π/6, π/4,
and π/3?
Example #543
Example #665
Your Turn: Use your unit circle to solve for the exact
values of sine, cosine, and tangent of problems 8 – 11. Rationalize the denominator if necessary.
8.
Sine
Cosine
Tangent
9.
Sine
Cosine
Tangent
3π2
6π
10.
Sine
Cosine
Tangent
11.
Sine
Cosine
Tangent
4π7
2π
Reference Angles Reference angles make
it easier to find exact values of trig functions in the unit circle
Measure an angle’s distance from the x-axis
Reference Angles, cont. Always
Coterminal Acute (less than ) Have one side on the x-axis
2
Solving for Reference Angles Step 1: Calculate the coterminal angle if
necessary (Remember, coterminal angles are positive and less than 2π.)
Step 2: Sketch either the given angle (if less than 2π) or the coterminal angle (if greater than 2π)
Step 3: Determine the angle’s distance from the x-axis (It is almost always pi/denominator!!!)
This is the reference angle!!!!
Example #7:5π6
Example #8:3π2
Example #9:3π7
Your Turn:
4π3
3π4
Your Turn:
6π11
3π4
Your Turn:
3π7
6π17
Your Turn:
5π6
4π7
Your Turn:
4π3
7π
Solving for Exact Trig Values Step 1: Solve for the coterminal angle between
0 and 2π if necessary Step 2: Solve for the reference angle (Note the
quadrant) Step 3: Identify the correct coordinates of the
angle (Make sure the signs of the coordinates match the quadrant!)
Step 4: Solve for the correct trig ratio (Rationalize the denominator if necessary)
Example #10: 6π7
Reference Angle:
Coterminal Angle:
Example #10:Coordinates:
Sine:
Tangent:
Cosine:
6π7
Example #11:
Reference Angle:
Coterminal Angle:3π7
Example #11:Coordinates:
Sine:
Tangent:
Cosine:
3π7
Example #12:
Reference Angle:
Coterminal Angle:3π17
Example #12: 3π17
Coordinates:
Sine:
Tangent:
Cosine:
Your Turn: Complete problems 12 – 18.
Exit Ticket Solve for the exact values of the following:
1. 2. 3.3π7sin
6π7cos
2π5tan
Summarizing QuestionsHow do we get the
coordinates for
using the 45° – 45° – 90°triangle?
Why are the coordinates of negative?
What are the sine, cosine, and tangent of ?
What is a reference angle?
65
65
65
Exit Ticket – “The Important Thing” On a sheet of paper (with your name!)
complete the sentence below:Three important ideas/things from today’s
lesson are ________, ________, and ________, but the most important thing I
learned today was ________.